The Szlenk index and local l1-indices
Abstract
We introduce two new local l1-indices of the same type as the Bourgain l1 index; the l1+-index and the l1+-weakly null index. We show that the l1+-weakly null index of a Banach space X is the same as the Szlenk index of X, provided X does not contain l1. The l1+-weakly null index has the same form as the Bourgain l1 index: if it is countable it must take values omegaalpha for some alpha<omega1. The different l1-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its local l1-index, via the l1+-weakly null index. We show that I(C(omegaomegaalpha))=omega1+alpha+1.
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